### Abstract

In two recent papers (Aizawa et al., 2013 [15]) and (Aizawa et al., 2015 [16]), representation theory of the centrally extended l-conformal Galilei algebra with half-integer l has been applied so as to construct second order differential equations exhibiting the corresponding group as kinematical symmetry. It was suggested to treat them as the Schrödinger equations which involve Hamiltonians describing dynamical systems without higher derivatives. The Hamiltonians possess two unusual features, however. First, they involve the standard kinetic term only for one degree of freedom, while the remaining variables provide contributions linear in momenta. This is typical for Ostrogradsky's canonical approach to the description of higher derivative systems. Second, the Hamiltonian in the second paper is not Hermitian in the conventional sense. In this work, we study the classical limit of the quantum Hamiltonians and demonstrate that the first of them is equivalent to the Hamiltonian describing free higher derivative nonrelativistic particles, while the second can be linked to the Pais-Uhlenbeck oscillator whose frequencies form the arithmetic sequence ω_{k}=(2k-1), k=1, . . ., n. We also confront the higher derivative models with a genuine second order system constructed in our recent work (Galajinsky and Masterov, 2013 [5]) which is discussed in detail for l=3/2.

Original language | English |
---|---|

Pages (from-to) | 244-254 |

Number of pages | 11 |

Journal | Nuclear Physics B |

Volume | 896 |

DOIs | |

Publication status | Published - 1 Jul 2015 |

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### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

*Nuclear Physics B*,

*896*, 244-254. https://doi.org/10.1016/j.nuclphysb.2015.04.024

**On dynamical realizations of l-conformal Galilei and Newton-Hooke algebras.** / Galajinsky, Anton; Masterov, Ivan Victorovich.

Research output: Contribution to journal › Article

*Nuclear Physics B*, vol. 896, pp. 244-254. https://doi.org/10.1016/j.nuclphysb.2015.04.024

}

TY - JOUR

T1 - On dynamical realizations of l-conformal Galilei and Newton-Hooke algebras

AU - Galajinsky, Anton

AU - Masterov, Ivan Victorovich

PY - 2015/7/1

Y1 - 2015/7/1

N2 - In two recent papers (Aizawa et al., 2013 [15]) and (Aizawa et al., 2015 [16]), representation theory of the centrally extended l-conformal Galilei algebra with half-integer l has been applied so as to construct second order differential equations exhibiting the corresponding group as kinematical symmetry. It was suggested to treat them as the Schrödinger equations which involve Hamiltonians describing dynamical systems without higher derivatives. The Hamiltonians possess two unusual features, however. First, they involve the standard kinetic term only for one degree of freedom, while the remaining variables provide contributions linear in momenta. This is typical for Ostrogradsky's canonical approach to the description of higher derivative systems. Second, the Hamiltonian in the second paper is not Hermitian in the conventional sense. In this work, we study the classical limit of the quantum Hamiltonians and demonstrate that the first of them is equivalent to the Hamiltonian describing free higher derivative nonrelativistic particles, while the second can be linked to the Pais-Uhlenbeck oscillator whose frequencies form the arithmetic sequence ωk=(2k-1), k=1, . . ., n. We also confront the higher derivative models with a genuine second order system constructed in our recent work (Galajinsky and Masterov, 2013 [5]) which is discussed in detail for l=3/2.

AB - In two recent papers (Aizawa et al., 2013 [15]) and (Aizawa et al., 2015 [16]), representation theory of the centrally extended l-conformal Galilei algebra with half-integer l has been applied so as to construct second order differential equations exhibiting the corresponding group as kinematical symmetry. It was suggested to treat them as the Schrödinger equations which involve Hamiltonians describing dynamical systems without higher derivatives. The Hamiltonians possess two unusual features, however. First, they involve the standard kinetic term only for one degree of freedom, while the remaining variables provide contributions linear in momenta. This is typical for Ostrogradsky's canonical approach to the description of higher derivative systems. Second, the Hamiltonian in the second paper is not Hermitian in the conventional sense. In this work, we study the classical limit of the quantum Hamiltonians and demonstrate that the first of them is equivalent to the Hamiltonian describing free higher derivative nonrelativistic particles, while the second can be linked to the Pais-Uhlenbeck oscillator whose frequencies form the arithmetic sequence ωk=(2k-1), k=1, . . ., n. We also confront the higher derivative models with a genuine second order system constructed in our recent work (Galajinsky and Masterov, 2013 [5]) which is discussed in detail for l=3/2.

UR - http://www.scopus.com/inward/record.url?scp=84946026686&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84946026686&partnerID=8YFLogxK

U2 - 10.1016/j.nuclphysb.2015.04.024

DO - 10.1016/j.nuclphysb.2015.04.024

M3 - Article

VL - 896

SP - 244

EP - 254

JO - Nuclear Physics B

JF - Nuclear Physics B

SN - 0550-3213

ER -